Artificial Intelligence and Soft Computing: Behavioral ... - Arteimi.info

c) ¬ ( ¬ P ) = P
d) ¬ ( ∃X P) = ∀ X ¬P
e) ¬ ( ∀ X P) = ∃ X ¬ P
In the present context, we rewrite the sentence as
∀ X (¬ Child (X) ∨ ¬ (∃Y (Takes (X, Y) ∧ Biscuit (Y))) ∨ Loves (john, X ))
⇒∀X (¬ Child (X) ∨ ∀ Y (¬ Takes (X, Y) ∨ ¬ Biscuit (Y)) ∨ Loves
(john, X))
Step III: Renaming the variables within the scope of quantifiers:
Rename ∃X by ∃ Y when {∃ X} is a subset / proper subset of {∀ X}. In
the present context, since X and Y are distinct, the above operation cannot be
carried out.
Step IV: Moving of quantifiers in the front of the expression: Bring all
quantifiers at the front of the expression.
Applying this on the example yields:
⇒∀ X ∀ Y ¬ Child (X) ∨ ¬ Takes (X,Y) ∨ ¬ Biscuit (Y) ∨ Loves (john, X)
Step V: Replacing existential quantifier as Skolem function of essential
quantifiers: When an existential quantifier (Y) precedes an essential quantifier
(X), replace Y as S (X), where S is the Skolem function [3]. In this example,
since Y is not a subset of X, such a situation does not arise. Also the essential
quantifier is dropped from the sentence.
Step VI: Putting the resulting expression in conjunctive normal form
(CNF): For example, if the original expression is in the form P ∨ (Q ∧ R),
then replace it by (P ∨ Q) ∧ ( P ∨ R).
In the present context, the resulting expression corresponding to
expression (3) being in CNF, we need not do any operation at this step.
Step VII: Writing one clause per line: If the original expression is of the
following CNF, then rewrite each clause/ line, as illustrated below.